# Derivative of An Integral

1. Jun 15, 2013

### Justabeginner

1. The problem statement, all variables and given/known data
Find the derivative of the function:

$f(x)= ∫e^sin(t) dt.$ (A is cos (x) and B is (x^2))

2. Relevant equations

3. The attempt at a solution

I read this site http://mathmistakes.info/facts/CalculusFacts/learn/doi/doi.html and I was wondering how I would be able to determine what problems this technique would work for. It says it has to be an open interval, and the function must be continuous. But in this case, since sin is defined only from -1 to 1, this would not work right? Then what must I do in such a case? Do I take the integral of the function and then plug in the upper and lower limits? I am utterly confused. Thank you.

Last edited: Jun 15, 2013
2. Jun 15, 2013

### HallsofIvy

Staff Emeritus
You mean $f(x)= \int e^{sin(t)}dt$. But what do "A" and "B" have to do with this?
And how does "x" come into it?

If you mean $f(x)= \int_a^x e^{sin(t)} dt$ then the derivative with respect to x is given by the "Fundamental Theorem of Calculus": The derivative, with respect to x, of $\int_a^x F(t)dt$ is F(x) no matter what a is.

3. Jun 15, 2013

### Justabeginner

A and B are the lower and upper limits of the integral but every time I put it in LaTex form it didn't pull up as I wanted it to, so I took it out. X is the variable used in the lower and upper limits of the integral.

So the integral limits are irrelevant to the problem itself, and by the Fundamental Theorem of Calculus, the derivative of ANY integral is F(x)?

Thank you.

4. Jun 15, 2013

### LCKurtz

He means$$f(x) = \int_{\cos x}^{x^2}e^{\sin t}\, dt$$

5. Jun 15, 2013

### Zondrina

This will allow you to quickly apply the fundamental theorem :

$$\frac{d}{dx} \int_{a(x)}^{b(x)} f(t) dt = f(b(x))(b'(x)) - f(a(x))(a'(x))$$

6. Jun 15, 2013

### Justabeginner

So in any question which asks me to find the derivative of an integral, this formula would be utilized?

[2x* e^ sin(x^2)] + [e^(sin(cos x)) * sin x] -Is this a correct application of the rule?

Thank you.

7. Jun 15, 2013

### QED Andrew

Yes, the formula posted by Zondrina is generally applicable. Furthermore, you correctly applied it to your problem. Good work! :-)

8. Jun 15, 2013

### Justabeginner

Thank you! :)